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Theorem bj-ceqsaltv 34100
Description: Version of bj-ceqsalt 34099 with a disjoint variable condition on 𝑥, 𝑉, removing dependency on df-sb 2061 and df-clab 2797. Prefer its use over bj-ceqsalt 34099 when sufficient (in particular when 𝑉 is substituted for V). (Contributed by BJ, 16-Jun-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-ceqsaltv ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝑉
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bj-ceqsaltv
StepHypRef Expression
1 bj-elissetv 34088 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
213anim3i 1146 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∃𝑥 𝑥 = 𝐴))
3 bj-ceqsalt0 34097 . 2 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ ∃𝑥 𝑥 = 𝐴) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
42, 3syl 17 1 ((Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) ∧ 𝐴𝑉) → (∀𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  w3a 1079  wal 1526   = wceq 1528  wex 1771  wnf 1775  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-12 2167
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-ex 1772  df-nf 1776  df-clel 2890
This theorem is referenced by:  bj-ceqsalgvALT  34105
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